Institute of Mathematics and Informatics Bulgarian Academy of Sciences

Citation:

Serdica Mathematical Journal, Vol. 28, No 4, (2002), 305p-328p

Abstract:

This paper is a survey which also contains some new results on
the nonlinear approximation with regard to a basis or, more generally, with
regard to a minimal system. Approximation takes place in a Banach or in
a quasi-Banach space. The last decade was very successful in studying nonlinear
approximation. This was motivated by numerous applications. Nonlinear
approximation is important in applications because of its increased
efficiency. Two types of nonlinear approximation are employed frequently
in applications. Adaptive methods are used in PDE solvers. The m-term
approximation considered here is used in image and signal processing as well
as the design of neural networks. The basic idea behind nonlinear approximation
is that the elements used in the approximation do not come from
a fixed linear space but are allowed to depend on the function being approximated.
The fundamental question of nonlinear approximation is how
to construct good methods (algorithms) of nonlinear approximation. In this
paper we discuss greedy type and thresholding type algorithms.

Description:

*This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003